Quads are great, and preferred when creating the mesh, as the tools like 'em better, and you don’t have all those extra edges cluttering up the place, but…
3a) Ultimately, EVERYTHING uses triangles as that is what hardware uses.
Game engines are no exception as they leverage the acceleration afforded by the hardware to achieve real-time frame-rates.
Newer commercial game engines happily accept quads, triangulate, calculate strip & fan primitives (used by hardware to accelerate rendering… & BTW, requires triangles 
) on the fly. And quad meshes typically work fine, not so much because tri’s are no longer important, but more so because these newer engines accept meshes with much higher poly counts so that the angles between edges are much less… and therefore the deformation is spread out across more faces, requiring each individual face to distort less during animation. Thus making it easier for each quad to maintain it’s initial “concavity” or “convex-ness”. (Now I’m makin’ up words 
)
However, if you are using an older, less advanced engine which requires much lighter meshes (i.e. fewer polys), tight control of the triangulated mesh is necessary. This is due to a quad having that ambiguous bisecting edge that turns it into 2 triangles. The game engine has no way of knowing which direction that edge bisects the quad unless you explicitly tell it.
Example:
Create a single quad plane.
Select 2 vertices on opposite corners and raise them up on Y axis.
Use the knife tool (K) to connect them, forming the “bisecting edge” that turns the quad into 2 tri’s.
In shaded mode, this will form a sort of “tent” shape. Whether that “tent” is right side up, or upside down depends entirely on the direction of this new edge. With the new edge selected, form the Ctrl-E edge menu, choose “Rotate edge CW”. Notice how this flips the “tent” to the opposite direction? This is the difference between a “concave” and “convex” quad… all dependent upon that bisecting edge.
This is why the previously stated view that quads “distort better than triangles during animation”, does not hold water for low poly game meshes. When you distort a mesh by animating it, whether a given quad becomes concave or convex is directly dependent upon the direction of the bisecting edge. So if you’re doing very low poly, you need to triangulate prior to export, and go thru your mesh, turning the new edges as required to try and maintain the same type of logical “edge flow” that modelers look to achieve with quad-only workflows, and preview your animations to see in any edges need turned in the area of the joints because they are “going concave” instead of convex or vice versa.
